Stellation

In geometry, the stellation is a process of construction new Polygone S (into two Dimension S), new Polyèdre S in three dimensions, or, in general, new Polytope S in N dimensions. The process consists in extending from the elements such as the plane edges or faces, generally in a symmetrical way, until each one of them meet again. The new figure is a stellation of the original.

The definition of Kepler

In 1619, Kepler defined the Stellation for the polygons and the polyhedrons, like the process of extension of the edges or the faces until they meet to form a new polygon or a new polyhedron. It étoila thus the Dodecahedron to obtain two of the regular spangled polyhedrons (two of the four solid of Kepler-Poinsot).

Spangled polygons

A stellation of a regular polygon is a spangled Polygone or a made up Polygone.

It can be represented by the symbol {n/m} , where N is the number of tops, and m , the not used in the sequence of the edges around him. If m is equal to one, it is a stellation zero , and a regular polygon {N} . And thus, (M-1) ^ère stellation is {n/m} .

A made up polygon appears if N and m have a common divider, and the whole stellation requires multiple cyclic ways to supplement it. For example, a hexagramme {6/2} is made by 2 triangles {3}, and {10/4} is made by 2 pentagrams {5/2}.

A regular n-gone has (n-4) /2 stellations if N is even, and (n-3) /2 stellations if N is odd.

Like the Heptagon, the Octogone has also two stellation octogrammic, one, {8/3} being a spangled Polygone, and the other, {8/2}, being the compound of two square.

Spangled polyhedrons

The stellation of the polyhedrons leads to the spangled polyhedral . The plane faces of a polyhedron divide space into many discrete cells. For a symmetrical polyhedron, these cells will form groups, or units, cells in conformity - we say that the cells in such units in conformity are in the same way standard. A common method to find stellation implies the selection of one or more types of cells.

This can lead to an enormous number of possible forms, therefore, more thorough criteria are often forced to reduce the whole of these stellations so that is significant and single in a certain manner.

A number of cells forming a layer closed around its core is called a shell. For a symmetrical polyhedron, a shell can be made up of one or more types of cells.

Based on such ideas, several interesting restrictive categories were identified.

Stellations principal line (hand-line). By adding successive shells to the core of the polyhedron, that led to the whole of the principal line stellations.
Stellations fully constant (Fully supported). the faces of the lower part of a cell can appear in an external way like a " surplomb". In a fully supported stellation, there do not exist such overhangs, and all the visible parts of a face are seen from the same with dimensions one.
Stellations monoacrales. Literally " point unique". Where there exists only one kind of point, or top, in a stellation (i.e all the tops are in conformity with a single symmetrical orbit), the stellation is monoacrale. All the stellations of this kind are fully constant.
Stellations primary educations Where a polyhedron has plans of a symmetry mirror, the edges falling into these plans are known as being placed in primary lines. If all the edges are placed in primary lines, the stellation is primary. All the primary stellations are fully constant.
Stellations of Miller In The Fifty-Nine Icosahedra Coxeter, Of the Valley, Flather and Petrie record five rules suggested by Miller. Although these rules precisely refer to the geometry of the Icosaèdre, they can be easily adapted to function with arbitrary polyhedrons. They ensure, among other things, that the rotational symmetry of the original polyhedron is preserved, and that each stellation is different in are external appearance. The four kind of stellation just definite are all of the subsets of the stellations of Miller.

We can also identify other categories:

a partial stellation , where one does not extend all the elements of a given dimensionnality.
a under-symmetrical stellation where all the elements symmetrically are not extended.

The solid of Archimedes and their duaux can also be spangled. Here, we generally add the following rule: the plane character of all the original faces must be present in the stellation, i.e we should not consider the partial stellations. For example the Cube is not regarded as a stellation of the Cuboctaèdre. There exists:

4 stellations of the rhombic Dodécaèdre
187 stellations of the Triakitétraèdre
358.833.097 stellations of the rhombic Triacontaèdre
17 stellations of the Cuboctaèdre (4 are shown in the " models of polyèdre" of Wenninger)
of the unknown stellations of the Icosidodécaèdre, but much more than above! (19 are shown in the list of the owners of polyhedron of Wenninger)

Seventeen of the not-convex uniform polyhedrons are stellations of solids of Archimedes.

Rules of Miller

With the rules of Miller, we find:

That there does not exist stellation of the Tétraèdre, because all the faces are adjacent.
That there does not exist stellation of the Cube, because the not-adjacent faces is parallel and thus, cannot be wide to meet on new edges.
That there exists 1 stellation of the Octaèdre, the spangled Octangle
Which there exist 3 stellations of the Dodécaèdre: the Small dodecahedron spangled, the Large dodecahedron and the Large dodecahedron spangled, they are all of the solid of Kepler-Poinsot.
That there exist 58 stellations of the Icosaèdre, including the Grand icosahedron (one of the solids of Kepler-Poinsot), the 2 {{E}} stellation and the final stellation of the icosahedron. The 59e model in The 59 Icosahedra is the original icosahedron itself.

Many " stellations of Miller" cannot be obtained directly by using the method of Kepler. For example, much have hollow centers where the original faces and the edges of the polyhedron core are entirely missing: it of left there nothing which can be spangled. This anomaly did not hold the attention until Inchbald (2002).

Other rules for the stellation

The rules of Miller represent the manner " by no means; correcte" to enumerate the stellations. They are based on the combination of parts in the diagram of stellation in certain manners, and do not take account of the topology of the resulting faces. Like such, there exist certain completely reasonable stellations of the icosahedron which do not form part of their list - one among it was identified by James Bridge in 1974, while some " stellations of Miller" are criticables as for knowing if it must be looked at as of the stellations - one of the icosahedral unit includes/understands several completely disconnected cells floating symmetrically in space.

Up to now, an alternative whole of rules which takes that in account was not fully developed. The majority of made progress are based on the concept stating that the stellation is the reciprocal process of the Facettage , by which parts can be removed polyhedron without creating new tops. For each stellation of a certain polyhedron, there exists a dual faceting of a dual Polyèdre, and vice versa. By studying facetings of dual, we gain in perspicacity in the stellations of the original. Bridge found its new stellation of the icosahedron by studying facetings of its dual, the dodecahedron.

Certain polyedrists adopts the point of view which the stellation is a process with two directions, such as two unspecified polyhedrons dividing the same plane faces are stellations one of the other. This is comprehensible if one designs a suitable general algorithm to be used in a computer program, but which is not differently of a great particular help.

Many examples of stellations can be found in the list of the models of stellations of Wenninger.

Nomenclature of the stellations

John Conway conceived a terminology for the Polygone S, the Polyèdre S and the spangled Polychore S (Coxeter 1974). In this system, the process of extension of edge to create a new figure is called stellation , which extends the faces is called widening (greatening) and what extends the cells is called aggrandissement (aggrandizement) (this last is not applied to the polyhedrons). This allows a systematic use of words such as “spangled”, “broad” and “large” by conceiving the names for the resulting figures. For example, Conway proposed some minor variations with the names of the polyhedral of Kepler-Poinsot.

See too

the List of the models of polyhedron of Wenninger: includes 44 forms spangled of octahedral, dodecahedron, icosahedron and icosidodécaèdre enumerated in the book " Polyhedron Models" of Magnus Wenninger published in 1974.
the polyhedric compounds: includes 5 regular compounds and 4 regular duaux compounds.

References

Bridge, NR. J.; Facetting the dodecahedron, Acta Crystallographica A30 (1974), pp. 548-552.
Coxeter, H.S.M.; Regular complex polytopes (1974).
Coxeter, H.S.M.; Valley, P.; Flather, H.T.; and Kneaded, J.F. The Fifty-Nine Icosahedra . Stradbroke, England: Tarquin Publications (1999).
Inchbald, G.; In search off the lost will icosahedra, The Mathematical Gazette 86 (2002), p.p. 208-215.
Messer, P.; Stellations off the rhombic triacontahedron and beyond, Symmetry: culture and science, 11 (2000), p 201-230.

External bonds

Stellation of the icosahedron and faceting of the dodecahedron
Stella: Polyhedron Navigator - Software for exploring will polyhedra and printing Nets for to their physical construction. Includes uniform will polyhedra, stellations, compounds, Johnson solids, etc
Enumeration off stellations
Bulatov, V. '' Polyhedra Stellation. ''
The Fifty-Nine Icosahedra - Applet
59 Stellations off the Icosahedron, George Binder
Stellation: Beautiful Maths

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